- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 425031, 10 pages

http://dx.doi.org/10.1155/2013/425031

## Pullback Attractors for Nonautonomous 2D-Navier-Stokes Models with Variable Delays

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received 5 January 2013; Accepted 30 March 2013

Academic Editor: de Dai

Copyright © 2013 Xiaoli Liu and Yejuan Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using a method based on the concept of the Kuratowski measure of the noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions, we prove the existence of a unique pullback attractor in higher regularity space for the multivalued process associated with the nonautonomous 2D-Navier-Stokes model with delays and without the uniqueness of solutions.

#### 1. Introduction

It is well known that the Navier-Stokes equations are very important in the understanding of fluids motion and turbulence. These equations have been studied extensively over the last decades (see [1–3], and the references cited therein). Recently, Caraballo and Real [4] considered global attractors for functional Navier-Stokes models with the uniqueness of solutions and for the delay, so that a wide range of hereditary characteristics (constant or variable delay, distributed delay, etc.) can be treated in a unified way. Very recently, Marín-Rubio and Real [5] used the theory of multivalued dynamical system to establish the existence of attractors for the 2D-Navier-Stokes model with delays, when the forcing term containing the delay is sublinear and only continuous.

For the study of asymptotic behavior for functional partial differential equations without the uniqueness of solutions, as far as we know, not many papers have been published. However, some results in the finite dimensional context can be found in [6, 7] (see also [8–10] for some preliminary and interesting results on the structure of the attractors for ordinary differential delay systems).

The pullback attractor is a possible approach to define an “attractor” for the nonautonomous dynamical systems, the long time behavior of nonautonomous dynamical systems is an interesting and challenging problem; see, for example, [11–19], and so forth. The purpose of our current paper is to study existence of pullback attractors for the following functional Navier-Stokes problem: where is an open bounded set with regular boundary , is the kinematic viscosity, is the velocity field of the fluid, is the pressure, is the initial time, is a nondelayed external force field, is another external force term and contains some memory effects during a fixed interval of time of length , is an adequate given delay function, and the initial datum on the interval .

Using the technique of measure of noncompactness, noting that all norms on finite dimensional spaces are equivalent, we apply the new method to check the pullback -limit compactness given in [20] and then get the existence of the pullback attractors in .

We consider the following usual abstract spaces: where the closure of in with norm and inner product , where for , where the closure of in with norm and associated scalar product , where for , Note that , where the injections are dense and compact. We will use for the norm in and for the duality pairing between and .

Define the trilinear form on by

Now, let us establish some assumptions for (1).

We assume that the given delay function satisfies , and there exists a constant satisfying Furthermore, we suppose that and satisfy the following assumptions:(H1) is measurable for all ,(H2) is continuous for all ,(H3) there exist positive constants , such that for any , (H4) there exists a fixed such that for any , the external force satisfies

Set as , by , for all . Denote by the corresponding orthogonal projection . We further set . The Stokes operator is self-adjoint and positive from to . The inverse operator is compact. Excluding the pressure, the system (1) can be written in the form

#### 2. Preliminaries

Let be a complete metric space with metric , and denote by the class of nonempty subsets of . As usual, let us denote by the Hausdorff semidistance between and , which are defined by where . Finally, denote by the open neighborhood of radius of a subset of a Banach space .

*Definition 1. *A family of mappings is called to be a multivalued process (MVP in short) if it satisfies(1)for all ;(2)for all .

Let be a nonempty class of parameterized sets .

*Definition 2. *Let be a multivalued process on . One says that is(1) pullback -dissipative, if there exists a family , so that for any and each , there exists a such that
(2) pullback -limit-set compact with respect to each , if for any and , there exists a such that
where is the Kuratowski measure of noncompactness.

*Definition 3. *A family of nonempty compact subsets is called to be a pullback -attractor for the multivalued process , if it satisfies(1) is invariant; that is,
(2) is pullback -attracting; that is, for every and any fixed ,

Let , be two Banach spaces, and let , be their dual spaces, respectively. We also assume that is a dense subspace of , the injection is continuous, and its adjoint is densely injective.

Theorem 4 (see [21, 22]). *Let be two Banach spaces satisfy the previous assumptions, and let be a multivalued process on and , respectively. Assume that is upper semicontinuous or weak upper semicontinuous on . If for fixed , maps compact subsets of into bounded subsets of , then is norm-to-weak upper semicontinuous on .*

By slightly modifying the arguments of Theorem 3.4 and Remark 3.9 in [21], we have the following.

Theorem 5. *Let be a Banach space, and let be a multivalued process on . Also let be norm-to-weak upper semicontinuous in for fixed , ; that is, if , then for any , there exist a subsequence and a such that (weak convergence). Then the multivalued process possesses a pullback -attractor in given by
**
if and only if is pullback -dissipative and pullback -limit-set compact with respect to each , where is pullback -absorbing for the multivalued process .*

A multivalued process is said to be pullback -asymptotically upper-semicompact in if for each fixed , any , any sequence with , with , and any with ; this last sequence is relatively compact in .

*Remark 6. *Let be a multivalued process on . Then is pullback -asymptotically upper-semicompact if and only if is pullback -limit-set compact; see [21].

Let be a Banach space, and let be a given positive number (the delay time). Denote by the Banach space endowed with the norm Let us consider a class of sets parameterized in time, . To study the pullback -limit-set compactness of the multivalued process on , we need the following result from [20].

Theorem 7. *Let be a multivalued process on . Suppose that for each , any and , there exist , a finite dimensional subspace of , and a such that*(1)* for each fixed ,
*(2)* for all , , with ,
*(3)* for all , ,
**where is the canonical projector. Then is pullback -limit-set compact in with respect to each .*

#### 3. Existence of an Absorbing Family of Sets in

By the classical Faedo-Galerkin scheme and compactness method, analogous to the arguments in [5], we have the following.

Theorem 8. *Let one consider , , and assume that satisfies the hypotheses (H1)–(H3). Then, for each ,*(a)

*there exists a weak solution to problem (9) satisfying*(b)

*if , then there exists a strong solution to problem (9); that is,*

Given and , for each , we denote by the function defined on by the relation , . We also denote and . Let be the arbitrary positive constants, which may be different from line to line and even in the same line.

Thanks to Theorem 8, we can define a multivalued process as

We first need a priori estimates for the solution of (9) in the space and a necessary bound on the term , which will be very useful in our analysis; it relates the absorption property for the multivalued process on .

Lemma 9. *In addition to the assumptions (H1)–(H4), assume that
*

*holds true. Then*

*provided that is small enough.*

*Proof. *By the energy inequality and the Poincaré inequality, we have
We fixed two positive parameters and to be chosen later on. Then by and Young’s inequality, we can deduce that
Therefore,
Let to be determined later on. Then it follows that

Integrating between and , we have

Let ; note that and for all . Hence,
Combining (30) and (31) together, we get
Let and using (23), so we can choose positive constants and small enough such that and (where is given in the assumption ). Then, it follows that
Setting now instead of (where ), multiplying by , it holds
Note that , thus the conclusion (24) follows immediately from (34).

Finally, we will obtain the bound on the term . It follows from (28) that
Integrating from to , we have
Similar to the arguments of (31), we can deduce that
Recall that and . By (24) and (36)-(37), we have (25) as desired, and thus the proof of this lemma is completed.

By slightly modifying the proof of Lemma 1.1 in [23], we have the following result.

Lemma 10. *
Let be given arbitrarily. Let , , and be three positive locally integrable functions on such that is locally integrable on , which satisfy that
**
where , , and are positive constants. Then
*

Now we state and prove the main result in this section.

Theorem 11. *Suppose in addition to the hypotheses in Lemma 9, assume that
**
holds true. Then the multivalued process on is pullback -dissipative.*

*Proof. *We take the inner product of (9) with , we obtain
Now we evaluate the terms, using and Young’s inequality, and we arrive to
Next,
Thanks to (41)–(43) and the fact that for , we can deduce that
and consequently,
Since and , it is easy to see that . Then

Let be given arbitrarily and taking such that . In order to apply Lemma 10, in view of (24), now we firstly obtain
Then, it follows from (24) and (25) that
Combining (25) and (47)-(48) together, by Lemma 10, we can conclude that
where
Therefore, if we take such that , then similar to the above mentioned, we get

We denote by the set of all functions such that
and denote by the class of all families such that , for some , where denotes the family of all nonempty subsets of and denotes the closed ball in centered at zero with radius .

Denote by the nonnegative number given for each by
and consider the family of closed balls in defined by
It is straightforward to check that , and moreover, by (51) and (52), the family of is pullback -absorbing for the multivalued process on .

The proof of Theorem 11 is completed.

#### 4. Existence of the Pullback Attractors in

Theorem 12. *Suppose in addition to the hypotheses in Theorem 11 that . Then there exists a unique pullback -attractor for the multivalued process in .*

*Proof. *Since is a continuous compact operator in , by the classical spectral theory, there exist a sequence ,
and a family of elements of which are orthonormal in such that
Let in and be an orthogonal projector.

Let , where and . We decompose (9) as follows:
We divide the proof into three steps.

(1) For every fixed , any and , we observe that for any with ,
Taking the inner product in of (57) with , we get
By and Young’s inequality, we have
To estimate , we recall some inequalities [19]:
and thus
Note that , and set . Then by Young’s inequality, we can deduce that
By (60)–(64) and Poincaré inequality, we obtain
Applying the Gronwall’s lemma in the interval , it yieldsLet be given arbitrarily. Note that , then we can take large enough such that for any fixed ,
Combining (67) and (68) together, we can get for large enough,
On the other hand, thanks to Lemma 9 and Theorem 11, we can deduce that when and are large enough,
Thanks to (69) and (70), it follows from (66) that when and are large enough,

(2) Now we consider the ordinary functional differential system (58) and check the condition (2) in Theorem 7. Note that . Without generality, we assume that with . Hence
Notice that
Then, it follows from , , and (24) that
Since and is fixed,
Equations (74)–(75) imply that the condition (2) in Theorem 7 is proved.

(3) Invoking Theorem 7, in view of the previous arguments and Theorem 11, we can see that the multivalued process is pullback -limit-set compact and pullback -dissipative in .

In order to get the existence of pullback -attractors, by the proof of Theorem 3.2 in [21], now we only need to show the negative invariance of , where
and is a pullback -absorbing set of in .

Let . Then there exist sequences ,, , and such that
On the other hand, for sufficiently large,
Then by the pullback -limit-set compactness of the multivalued process , there is a subsequence of , which we still relabel as such that and
Clearly, .

We observe that is bounded in for sufficiently large. Then by slightly modifying the proof of the existence of solutions (see [16] for details), in view of Theorem 2.11 in [21], we can see that
This together with (77)–(79), we can deduce that , and thus the proof of Theorem 12 is finished.

#### Acknowledgments

This research was supported by the National Natural Science Foundation of China under Grant no. 10801066 and the Fundamental Research Funds for the Central Universities under Grant no. lzujbky-2011-47 and no. lzujbky-2012-k26. The Project was sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

#### References

- P. Constantin and C. Foias,
*Navier-Stokes Equations*, The University of Chicago Press, Chicago, Ill, USA, 1988. View at Zentralblatt MATH · View at MathSciNet - P. Marín-Rubio and J. C. Robinson, “Attractors for the stochastic 3D Navier-Stokes equations,”
*Stochastics and Dynamics*, vol. 3, no. 3, pp. 279–297, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Temam,
*Navier-Stokes Equations: Theory and Numerical Analysis*, vol. 2, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1979. View at MathSciNet - T. Caraballo and J. Real, “Attractors for 2D-Navier-Stokes models with delays,”
*Journal of Differential Equations*, vol. 205, no. 2, pp. 271–297, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Marín-Rubio and J. Real, “Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators,”
*Discrete and Continuous Dynamical Systems A*, vol. 26, no. 3, pp. 989–1006, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Caraballo, P. Marín-Rubio, and J. Valero, “Autonomous and non-autonomous attractors for differential equations with delays,”
*Journal of Differential Equations*, vol. 208, no. 1, pp. 9–41, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Caraballo, P. Marín-Rubio, and J. Valero, “Attractors for differential equations with unbounded delays,”
*Journal of Differential Equations*, vol. 239, no. 2, pp. 311–342, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. K. Hale and S. M. Verduyn-Lunel,
*Introduction to Functional Differential Equations*, Springer, Berlin, Germany, 1993. View at MathSciNet - J. Mallet-Paret and G. R. Sell, “Systems of differential delay equations: floquet multipliers and discrete Lyapunov functions,”
*Journal of Differential Equations*, vol. 125, no. 2, pp. 385–440, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Mallet-Paret and G. R. Sell, “The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay,”
*Journal of Differential Equations*, vol. 125, no. 2, pp. 441–489, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß, and J. Valero, “Non-autonomous and random attractors for delay random semilinear equations without uniqueness,”
*Discrete and Continuous Dynamical Systems A*, vol. 21, no. 2, pp. 415–443, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Caraballo and P. E. Kloeden, “Non-autonomous attractor for integro-differential evolution equations,”
*Discrete and Continuous Dynamical Systems S*, vol. 2, no. 1, pp. 17–36, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - T. Caraballo, P. E. Kloeden, and J. Real, “Pullback and forward attractors for a damped wave equation with delays,”
*Stochastics and Dynamics*, vol. 4, no. 3, pp. 405–423, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Caraballo, G. Łukaszewicz, and J. Real, “Pullback attractors for asymptotically compact non-autonomous dynamical systems,”
*Nonlinear Analysis: Theory, Methods and Applications A*, vol. 64, no. 3, pp. 484–498, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. N. Cheban, P. E. Kloeden, and B. Schmalfuß, “The relationship between pullback, forward and global attractors of nonautonomous dynamical systems,”
*Nonlinear Dynamics and Systems Theory*, vol. 2, no. 2, pp. 9–28, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. V. Chepyzhov and M. I. Vishik,
*Attractors for Equations of Mathematical Physics*, vol. 49, American Mathematical Society, Providence, RI, USA, 2002. View at MathSciNet - P. E. Kloeden and B. Schmalfuß, “Asymptotic behaviour of nonautonomous difference inclusions,”
*Systems & Control Letters*, vol. 33, no. 4, pp. 275–280, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. E. Kloeden and J. A. Langa, “Flattening, squeezing and the existence of random attractors,”
*Proceedings of the Royal Society of London A*, vol. 463, no. 2077, pp. 163–181, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Wang, C. Zhong, and S. Zhou, “Pullback attractors of nonautonomous dynamical systems,”
*Discrete and Continuous Dynamical Systems A*, vol. 16, no. 3, pp. 587–614, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. J. Wang and P. E. Kloeden, “The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain,”
*Discrete and Continuous Dynamical Systems A*. In press. - Y. Wang and S. Zhou, “Kernel sections on multi-valued processes with application to the nonlinear reaction-diffusion equations in unbounded domains,”
*Quarterly of Applied Mathematics*, vol. 67, no. 2, pp. 343–378, 2009. View at Google Scholar · View at MathSciNet - C.-K. Zhong, M.-H. Yang, and C.-Y. Sun, “The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations,”
*Journal of Differential Equations*, vol. 223, no. 2, pp. 367–399, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Temam,
*Infinite Dimensional Dynamical Systems in Mechanics and Physics*, Springer, New York, NY, USA, 1988. View at Publisher · View at Google Scholar · View at MathSciNet